= Quantum state ray
{title2=$[\psi]=\{\lambda|\psi\rangle:\lambda\in\mathbb C\setminus\{0\}\}$}
A <quantum state ray> is the <equivalence class> of nonzero <vectors> in a complex <Hilbert space> under multiplication by a nonzero <complex number>. Its span is a one-dimensional <linear subspace>, but the zero <vector> is not a state representative. A <unit vector> fixes the <norm> and leaves a <global phase> free. <Foundations I: States and Ensembles by John Preskill>[Preskill, Foundations I: States and Ensembles, §2.1, printed p. 2, ray equivalence and normalized representatives]{ref}
For a nonzero representative $|v\rangle$, the associated <pure state> <density operator> is
$$
\rho_{[v]}=\frac{|v\rangle\langle v|}{\langle v|v\rangle}.
$$
Replacing $|v\rangle$ by $\lambda|v\rangle$ multiplies both numerator and denominator by $|\lambda|^2$, so the quotient is unchanged. In particular, for every <POVM> effect $E$,
$$
\langle e^{i\gamma}\psi|E|e^{i\gamma}\psi\rangle
=e^{-i\gamma}e^{i\gamma}\langle\psi|E|\psi\rangle
=\langle\psi|E|\psi\rangle.
$$
This proves that a <global phase> changes no measurement <probability>. Changing the relative <quantum phase> of two nonzero components can change the <density operator>, as the following comparison shows.
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